Hall effects and weak localization in strong SO coupled systems : merging Keldysh, Kubo, and Boltzmann approaches via the chiral basis. SPIE, San Diego, August 12 th 2008 JAIRO SINOVA Texas A&M Univ. and Inst. Phys. ASCR Research fueled by: SWAN-NRI Branislav Nikolic U. of Delaware Allan MacDonald U of Texas Tomas Jungwirth Inst. of Phys. ASCR U. of Nottingham Joerg Wunderlich Cambridge-Hitachi Laurens Molenkamp Wuerzburg Kentaro Nomura U. Of Texas Ewelina Hankiewicz U. of Missouri Texas A&M U. Mario Borunda Texas A&M U. Nikolai Sinitsyn LANL Xin Liu Texas A&M U. Alexey Kovalev Texas A&M U. Brian Gallagher Other collaborators: Bernd Kästner, Satofumi Souma, Liviu Zarbo, Dimitri Culcer, Qian Niu, S-Q Shen,,Tom Fox, Richard Campton, Artem Abanov

I.Anomalous Hall Effect 1.History, semi-classical mechanism 2.Microscopic approach, IAHE 3.Merging the different linear theories a.AHE in graphene b.AHE in 2DEG+Rashba II.Spin Hall Effect 1.Spin accumulation with strong SO III.Weak Localization in GaMnAs 1.The experimental observations 2.Theory results

Anomalous Hall effect: Simple electrical measurement of magnetization like-spin Spin-orbit coupling “force” deflects like-spin particles I _ F SO _ _ _ majority minority V InMnAs controversial theoretically: three contributions to the AHE (intrinsic deflection, skew scattering, side jump scattering)

Intrinsic deflection Electrons have an “anomalous” velocity perpendicular to the electric field related to their Berry’s phase curvature which is nonzero when they have spin-orbit coupling. Electrons deflect to the right or to the left as they are accelerated by an electric field ONLY because of the spin-orbit coupling in the periodic potential (electronics structure) E Electrons deflect first to one side due to the field created by the impurity and deflect back when they leave the impurity since the field is opposite resulting in a side step. Related to the intrinsic effect: analogy to refraction from an imbedded medium Side jump scattering Skew scattering Asymmetric scattering due to the spin- orbit coupling of the electron or the impurity. This is also known as Mott scattering used to polarize beams of particles in accelerators.

(thanks to P. Bruno– CESAM talk) A history of controversy

COLLINEAR MAGNETIZATION AND SPIN-ORBIT COUPLING vs. CHIRAL MAGNET STRUCTURES AHE is present when SO coupling and/or non-trivial spatially varying magnetization (even if zero in average) SO coupled chiral states: disorder and electric fields lead to AHE/SHE through both intrinsic and extrinsic contributions Spatial dependent magnetization: also can lead to AHE. A local transformation to the magnetization direction leads to a non- abelian gauge field, i.e. effective SO coupling (chiral magnets), which mimics the collinear+SO effective Hamiltonian in the adiabatic approximation So far one or the other have been considered but not both together, in the following we consider only collinear magnetization + SO coupling

 Need to match the Kubo, Boltzmann, and Keldysh  Kubo: systematic formalism  Boltzmann: easy physical interpretation of different contributions (used to define them)  Keldysh approach: also a systematic kinetic equation approach (equivelnt to Kubo in the linear regime). In the quasiparticle limit it must yield Boltzmann eq. Microscopic vs. Semiclassical

CONTRIBUTIONS TO THE AHE: MICROSCOPIC KUBO APPROACH Skew scattering “side-jump scattering” Intrinsic AHE: accelerating between scatterings Skew σ H Skew  (  skew ) -1  2~ σ 0 S where S = Q(k,p)/Q(p,k) – 1~ V 0 Im[ ] Vertex Corrections  σ Intrinsic Intrinsic  σ 0 /ε F  n, q m, p n’, k n, q n’  n, q

FOCUS ON INTRINSIC AHE (early 2000’s): semiclassical and Kubo K. Ohgushi, et al PRB 62, R6065 (2000); T. Jungwirth et al PRL 88, 7208 (2002); T. Jungwirth et al. Appl. Phys. Lett. 83, 320 (2003); M. Onoda et al J. Phys. Soc. Jpn. 71, 19 (2002); Z. Fang, et al, Science 302, 92 (2003). Semiclassical approach in the “clean limit” Kubo: n, q n’  n, q STRATEGY: compute this contribution in strongly SO coupled ferromagnets and compare to experimental results, does it work?

Success of intrinsic AHE approach in comparing to experiment: phenomenological “proof” DMS systems (Jungwirth et al PRL 2002, APL 03) Fe (Yao et al PRL 04) layered 2D ferromagnets such as SrRuO3 and pyrochlore ferromagnets [Onoda and Nagaosa, J. Phys. Soc. Jap. 71, 19 (2001),Taguchi et al., Science 291, 2573 (2001), Fang et al Science 302, 92 (2003), Shindou and Nagaosa, Phys. Rev. Lett. 87, (2001)] colossal magnetoresistance of manganites, Ye et~al Phys. Rev. Lett. 83, 3737 (1999). CuCrSeBr compounts, Lee et al, Science 303, 1647 (2004) Berry’s phase based AHE effect is quantitative- successful in many instances BUT still not a theory that treats systematically intrinsic and extrinsic contribution in an equal footing Experiment  AH  1000 (  cm) -1 Theroy  AH  750 (  cm) -1 AHE in GaMnAs AHE in Fe

INTRINSIC+EXTRINSIC: REACHING THE END OF A 50 YEAR OLD DEBATE AHE in Rashba systems with weak disorder: Dugaev et al (PRB 05) Dugaev et al (PRB 05) Sinitsyn et al (PRB 05, PRB 07) Sinitsyn et al (PRB 05, PRB 07) Inoue et al (PRL 06) Inoue et al (PRL 06) Onoda et al (PRL 06, PRB 08) Onoda et al (PRL 06, PRB 08) Borunda et al (PRL 07), Nuner et al (PRB 07, PRL 08) Kovalev et al (PRB 08) All are done using same or equivalent linear response formulation–different or not obviously equivalent answers!!! The only way to create consensus is to show (IN DETAIL) agreement between ALL the different equivalent linear response theories both in AHE and SHE and THEN test it experimentally

Semiclassical Boltzmann equation Golden rule: J. Smit (1956): Skew Scattering In metallic regime: Kubo-Streda formula summary Calculation done easiest in normal spin basis

Golden Rule: Coordinate shift: Modified Boltzmann Equation: Berry curvature: velocity:current: Semiclassical approach II: Sinitsyn et al PRB 06

In metallic regime: Kubo-Streda formula: “AHE” in graphene Sinitsyn et al PRB 07

Comparing Botlzmann to Kubo in the chiral basis Sinitsyn et al PRB 07

A more realistic test AHE in Rashba 2D system Inversion symmetry  no R-SO Broken inversion symmetry  R-SO Bychkov and Rashba (1984) (differences: spin is a non-conserved quantity, define spin current as the gradient term of the continuity equation. Spin-Hall conductivity: linear response of this operator) n, q n’  n, q

AHE in Rashba 2D system Kubo and semiclassical approach approach: (Nuner et al PRB08, Borunda et al PRL 07) Only when ONE both sub-band there is a significant contribution When both subbands are occupied there is additional higher order vertex corrections that contribute

AHE in Rashba 2D system When both subbands are occupied the skew scattering is only obtained at higher Born approximation order AND the extrinsic contribution is unique (a hybrid between skew and side-jump) Kovalev et al PRB Rapids 08 Keldysh and Kubo match analytically in the metallic limit Numerical Keldysh approach (Onoda et al PRL 07, PRB 08) Solved within the self consistent T-matrix approximation for the self-energy

Testing the theory: in progress

AHE in Rashba 2D system: “dirty” metal limit? Is it real? Is it justified? Is it “selective” data chosing? Can the kinetic metal theory be justified when disorder is larger than any other scale? Onoda et al 2008

Spin Hall effect like-spin Take now a PARAMAGNET instead of a FERROMAGNET: Spin-orbit coupling “force” deflects like-spin particles I _ F SO _ _ _ V=0 non-magnetic Spin-current generation in non-magnetic systems without applying external magnetic fields Spin accumulation without charge accumulation excludes simple electrical detection Carriers with same charge but opposite spin are deflected by the spin-orbit coupling to opposite sides.

Spin Hall Effect (Dyaknov and Perel) Interband Coherent Response  (E F  ) 0 Occupation # Response `Skew Scattering‘  (e 2 /h) k F (E F  ) 1 X `Skewness’ [Hirsch, S.F. Zhang] Intrinsic `Berry Phase’  (e 2 /h) k F  [Murakami et al, Sinova et al] Influence of Disorder `Side Jump’’ [Inoue et al, Misckenko et al, Chalaev et al…] Paramagnets

First experimental observations at the end of 2004 Wunderlich, Kästner, Sinova, Jungwirth, cond-mat/ PRL 05 Experimental observation of the spin-Hall effect in a two dimensional spin-orbit coupled semiconductor system Co-planar spin LED in GaAs 2D hole gas: ~1% polarization CP [%] Light frequency (eV) Kato, Myars, Gossard, Awschalom, Science Nov 04 Observation of the spin Hall effect bulk in semiconductors Local Kerr effect in n-type GaAs and InGaAs: ~0.03% polarization (weaker SO-coupling, stronger disorder)

OTHER RECENT EXPERIMENTS “demonstrate that the observed spin accumulation is due to a transverse bulk electron spin current” Sih et al, Nature 05, PRL 05 Valenzuela and Tinkham cond- mat/ , Nature 06 Transport observation of the SHE by spin injection!! Saitoh et al APL 06 Room temperature SHE in ZnSe ??? Stern et al 06 (signal same as GaAs but SO smaller????)

The challenge: understanding spin accumulation in strongly spin-orbit coupled systems Spin is not conserved; analogy with e-h system Burkov et al. PRB 70 (2004) Spin diffusion length Quasi-equilibrium Parallel conduction Spin Accumulation – Weak SO

Spin Accumulation – Strong SO Mean Free Path? Spin Precession Length ?

SPIN ACCUMULATION IN 2DHG: EXACT DIAGONALIZATION STUDIES  so >>ħ/  Width>>mean free path Nomura, Wundrelich et al PRB 06 Key length: spin precession length!! Independent of  !!

1.5  m channel n n p y x z LED  m channel SHE experiment in GaAs/AlGaAs 2DHG - shows the basic SHE symmetries - edge polarizations can be separated over large distances with no significant effect on the magnitude - 1-2% polarization over detection length of ~100nm consistent with theory prediction (8% over 10nm accumulation length) Wunderlich, Kaestner, Sinova, Jungwirth, Phys. Rev. Lett. '05 Nomura, Wunderlich, Sinova, Kaestner, MacDonald, Jungwirth, Phys. Rev. B '05

H-bar for detection of Spin-Hall-Effect (electrical detection through inverse SHE) E.M. Hankiewicz et al., PRB 70, R (2004) Charge based measurements of ISHE (Numerical Keldysh calculation: no SO in leads)

Mesoscopic electron SHE L L/6 L/2 calculated voltage signal for electrons (Hankiewicz and Sinova)

Mesoscopic hole SHE L calculated voltage signal (Hankiweicz, Sinova, & Molenkamp) L L/ 6 L/2

New (smaller) sample 1  m 200 nm sample layout Experiments by Laruens Molenkamp group

SHE-Measurement

SUMMARY (AHE AND SHE) All linear theories treating disorder and non-trivial band structure have been merged in agreement Clear identification of semi-clasical contributions from the microscopic theory Many strongly spin-orbit coupled systems are dominated by the intrinsic contribution: old side-jump+intrinsic cancellations were an artifact of simple band structure (e.g. constant Berry curvature) Intrinsic SHE can also be observed in strongly spin-orbit coupled system with the induced spin-accumulation length scale in agreement with theory Charge based detection of intrinsic SHE seen in inverted semiconductor systems SWAN-NRI

Weak Localization in GaMnAs Quantum driven localization of time reversed paths interference. Each spin channel adds to the localization. In the presence of spin-orbit coupling one decouples channels in total angular momentum states. Singlet (zero total spin) is the one not affected BUT contributes with a negative sign to diffusion, i.e. Weak Antilocalization. e.g. 2D

Matsukura et al Physica E 2004 Kawabata A., Solid State Commun. 34 (1980) 432 Weak Localization at high magnetic fields Low field MR dominated by complicated AMR effects

Focus is on low magnetic field region where AMR dominates Rely on subtracting e-e interaction contribution which they attribute to the 1-D theory proportional to T -1/2. However they ignore that e-e contribution depends on the conductivity and strong AMR contributions will influence it. 1-D dimensionality is not quite justified given the length scales at the temperatures considered L so seems too large to have real meaning. For a strongly spin-orbit coupled system is should be lower. High field contribution ignored

B2?B2?

Rokhinson et al observed a ~1% negative MR at low temperature in a GaMnAs film which saturates at ~20mT and “states” that it is isotropic in field (ignoring the clear AMR in the data). The magnitude of the WL is stronger than the largest expected from the simples theory. One expects saturation at very large fields, not present in their experiment But they still ascribe this feature to weak localisation and furthermore argue that the presence of weak localisation is incompatible with the Fermi level being in strongly spin orbit coupled valence band ??!! ? ? ? Ferromagnetic transition temperatures   Magneto-crystalline anisotropy and coercively   Domain structure   Anisotropic magneto-resistance   Anomalous Hall effect   MO in the visible range   Non-Drude peak in longitudinal ac-conductivity  Ferromagnetic resonance  Domain wall resistance  TAMR  Success of metallic disorder valence band theory seems unimportant But is their main basis even right?

Theory of WL in GaMnAs Unlike the case of time-reversal symmetric systems there are no obvious Invariant representation when the energy scales are similar (exchange field, disorder, spin-orbit coupling, etc.)

Key result: for typical doping values and disorder WL is present!!! The main point is b/c disorder affects most the inter-band correlations which in the case of GaMnAs dominates the WAL contribution so the cross over from WAL to WL occurs before Eso is of the order of exchange energy.

SUMMARY (Weak Localization in GaMnAs) Interpretation of low magnetic field MR effects do not support a clear signature of WL (or WAL). Complicated AMR effects need to be taken into account more carefully For moderate Mn doping GaMnAs should show WL due to the large disorder scattering which limits the WAL corrections coming from interband correlations Interpretation of WL-> impurity band has no basis since the presence of SO coupling in the model does not create a WAL regime for moderate Mn doping!!! Effects of e-e interactions at low fields should incorporate AMR effects to correctly analyze the data

Spin-orbit coupling interaction (one of the few echoes of relativistic physics in the solid state) Ingredients: -“Impurity” potential V(r) - Motion of an electron Produces an electric field In the rest frame of an electron the electric field generates and effective magnetic field This gives an effective interaction with the electron’s magnetic moment CONSEQUENCES If part of the full Hamiltonian quantization axis of the spin now depends on the momentum of the electron !! If treated as scattering the electron gets scattered to the left or to the right depending on its spin!!

Non-equilibrium Green’s function formalism (Keldysh-LB) Advantages: No worries about spin-current definition. Defined in leads where SO=0 Well established formalism valid in linear and nonlinear regime Easy to see what is going on locally Fermi surface transport 3. Charge based measurements of SHE

PRL 05

INTRINSIC SPIN-HALL EFFECT: Murakami et al Science 2003 (cond-mat/ ) Sinova et al PRL 2004 (cont-mat/ ) as there is an intrinsic AHE (e.g. Diluted magnetic semiconductors), there should be an intrinsic spin-Hall effect!!! Inversion symmetry  no R-SO Broken inversion symmetry  R-SO Bychkov and Rashba (1984) (differences: spin is a non-conserved quantity, define spin current as the gradient term of the continuity equation. Spin-Hall conductivity: linear response of this operator) n, q n’  n, q

‘Universal’ spin-Hall conductivity Color plot of spin-Hall conductivity: yellow=e/8π and red=0 n, q n’  n, q

Disorder effects: beyond the finite lifetime approximation for Rashba 2DEG Question: Are there any other major effects beyond the finite life time broadening? Does side jump contribute significantly? Ladder partial sum vertex correction: Inoue et al PRB 04 Dimitrova et al PRB 05 Raimondi et al PRB 04 Mishchenko et al PRL 04 Loss et al, PRB 05 the vertex corrections are zero for 3D hole systems (Murakami 04) and 2DHG (Bernevig and Zhang 05) n, q n’  n, q + +…=0 For the Rashba example the side jump contribution cancels the intrinsic contribution!!

SHE conductivity: all contributions– Kubo formalism perturbation theory Skew  σ 0 S Vertex Corrections  σ Intrinsic Intrinsic  σ 0 /ε F  n, q n’  n, q = j = -e v = j z = {v,s z }

Anomalous Hall effect: what is necessary to see the effects? I _ F SO _ _ _ majority minority V Necessary condition for AHE: TIME REVERSAL SYMMETRY MUST BE BROKEN Need a magnetic field and/or magnetic order BUT IS IT SUFFICIENT? (P. Bruno– CESAM 2005)

Local time reversal symmetry being broken does not always mean AHE present Staggered flux with zero average flux: -- --  -- Is  xy zero or non-zero?  -- -- -- Translational invariant so  xy =0 Similar argument follows for antiferromagnetic ordering Does zero average flux necessary mean zero  xy ? -- -- -- 33 -- -- -- 33 No!! (Haldane, PRL 88) (P. Bruno– CESAM 2005)

Is non-zero collinear magnetization sufficient? (P. Bruno– CESAM 2005) In the absence of spin-orbit coupling a spin rotation of  restores TR symmetry and  xy =0 If spin-orbit coupling is present there is no invariance under spin rotation and  xy ≠0

(P. Bruno– CESAM July 2005) Collinear magnetization AND spin-orbit coupling → AHE Does this mean that without spin-orbit coupling one cannot get AHE? Even non-zero magnetization is not a necessary condition No!! A non-trivial chiral magnetic structure WILL give AHE even without spin-orbit coupling M x =M y =M z =0  xy ≠0 Bruno et al PRL 04